49 research outputs found

    Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures

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    AbstractThe AutoGraphiX system determines classes of extremal or near-extremal graphs with a variable neighborhood search heuristic. From these, conjectures may be deduced interactively. Three methods, a numerical, a geometric and an algebraic one are proposed to automate also this last step. This leads to automated deduction of previous conjectures, strengthening of a series of conjectures from Graffiti and obtention of several new conjectures, four of which are proved

    On some interconnections between combinatorial optimization and extremal graph theory

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    The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions

    Automated conjecturing III : property-relations conjectures

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    Discovery in mathematics is a prototypical intelligent behavior, and an early and continuing goal of artificial intelligence research. We present a heuristic for producing mathematical conjectures of a certain typical form and demonstrate its utility. Our program conjectures relations that hold between properties of objects (property-relation conjectures). These objects can be of a wide variety of types. The statements are true for all objects known to the program, and are the simplest statements which are true of all these objects. The examples here include new conjectures for the hamiltonicity of a graph, a well-studied property of graphs. While our motivation and experiments have been to produce mathematical conjectures-and to contribute to mathematical research-other kinds of interesting property-relation conjectures can be imagined, and this research may be more generally applicable to the development of intelligent machinery

    Graph Classes (Dis)satisfying the Zagreb Indices Inequality

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    International audience{Recently Hansen and Vukicevic proved that the inequality M1/n≀M2/mM_1/n \leq M_2/m, where M1M_1 and M2M_2 are the first and second Zagreb indices, holds for chemical graphs, and Vukicevic and Graovac proved that this also holds for trees. In both works is given a distinct counterexample for which this inequality is false in general. Here, we present some classes of graphs with prescribed degrees, that satisfy M1/n≀M2/mM_1/n \leq M_2/m: Namely every graph GG whose degrees of vertices are in the interval [c;c+c][c; c + \sqrt c] for some integer cc satisies this inequality. In addition, we prove that for any Δ≄5\Delta \geq 5, there is an infinite family of graphs of maximum degree Δ\Delta such that the inequality is false. Moreover, an alternative and slightly shorter proof for trees is presented, as well\ as for unicyclic graphs

    Learning-powered computer-assisted counterexample search

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    Treballs Finals de Grau de MatemĂ tiques, Facultat de MatemĂ tiques, Universitat de Barcelona, Any: 2023, Director: Kolja Knauer[en] This thesis explores the great potential of computer-assisted proofs in the advancement of mathematical knowledge, with a special focus on using computers to refute conjectures by finding counterexamples, sometimes a humanly impossible task. In recent years, mathematicians have become more aware that machine learning techniques can be extremely helpful for finding counterexamples to conjectures in a more efficient way than by using exhaustive search methods. In this thesis we do not only present the theoretical background behind some of these methods but also implement them to try to refute some graph theory conjectures

    Linear inequalities among graph invariants: Using GraPHedron to uncover optimal relationships

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    Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a "small" number of vertices. It greatly helps in conjecturing optimal linear inequalities, which are then hopefully proved for any number of vertices. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities for the stability number and the number of edges are obtained. To this aim, a problem of Ore (1962) related to the Turån Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter. © 2008 Wiley Periodicals, Inc

    On the Nullity Number of Graphs

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    The paper discusses bounds on the nullity number of graphs. It is proved in [B. Cheng and B. Liu, On the nullity of graphs. Electron. J. Linear Algebra 16 (2007) 60--67] that η≀n−D\eta \le n - D, where η\eta, n and D denote the nullity number, the order and the diameter of a connected graph, respectively. We first give a necessary condition on the extremal graphs corresponding to that bound, and then we strengthen the bound itself using the maximum clique number. In addition, we prove bounds on the nullity using the number of pendant neighbors in a graph. One of those bounds is an improvement of a known bound involving the domination number
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